Restaurant Señor de Huanca on the main plaza is a reliable choice for a cheap set lunch. For around one or two dollars you get soup, a main (typically rice, potatoes, and some form of meat), and sometimes a small dessert. The food is hearty and suited to the cold mountain air. The plaza location makes it easy to find.
Local specialities worth trying anywhere in Chivay include cuy (guinea pig, typically baked or fried), rocoto relleno (a spiced stuffed pepper from the Arequipa tradition), and sopa de quinoa. Most restaurants around the plaza serve similar set menus at comparable prices.